Question: What is the inverse of the function $f(x)=\dfrac{5x+2}{x-3}$ ? $f^{-1}(x) =$
Answer: Let's start by replacing $f(x)$ with $y$. $y=\dfrac{5x+2}{x-3}$ Now let's swap $x$ and $y$ and solve for $y$. $\dfrac{5y+2}{y-3}=x$ [Why do we swap x and y?] $\begin{aligned} \dfrac{5y+2}{y-3}&=x \\\\ 5y+2&=x(y-3) \\\\ 5y+2&=xy-3x \\\\ 5y-xy&=-3x-2 \\\\ y(5-x)&=-3x-2 \\\\ y&=\dfrac{-3x-2}{5-x} \end{aligned}$ In conclusion, this is the inverse function: $f^{-1}(x)=\dfrac{-3x-2}{5-x}$ [I saw someone solve this problem by originally solving for x. Were they wrong?]